grandes-ecoles 2014 QII.A.2

grandes-ecoles · France · centrale-maths1__psi Polar coordinates

Let $z \in \mathbb{C}$. We denote $C_z$ (respectively $\Omega_z$) the set of points in the plane with complex affixe $Z$ such that $|Z(Z-2z)| = 1$ (respectively $|Z(Z-2z)| < 1$). In this question we assume that $z$ is a real number denoted $a$. The curve $C_a$ in polar coordinates $(\rho, \theta)$ in the frame $\mathcal{R}'$ satisfies $$\left(\rho^2 + a^2\right)^2 - 4a^2 \rho^2 \cos^2\theta = 1$$ Simplify this equation when $a = 1$. Study and sketch the shape of the curve $C_1$.

Let $z \in \mathbb{C}$. We denote $C_z$ (respectively $\Omega_z$) the set of points in the plane with complex affixe $Z$ such that $|Z(Z-2z)| = 1$ (respectively $|Z(Z-2z)| < 1$).\\
In this question we assume that $z$ is a real number denoted $a$. The curve $C_a$ in polar coordinates $(\rho, \theta)$ in the frame $\mathcal{R}'$ satisfies
$$\left(\rho^2 + a^2\right)^2 - 4a^2 \rho^2 \cos^2\theta = 1$$
Simplify this equation when $a = 1$. Study and sketch the shape of the curve $C_1$.

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.A.4 QI.A.5 QI.A.6 QI.A.7 QI.B.1 QI.B.2 QI.B.3 QI.B.4 QII.A.1 QII.A.2 QII.B.1 QII.B.2 QII.C.1 QII.C.2 QII.C.3 QII.C.4 QII.C.5 QII.C.6 QII.C.7 QIII.A.1 QIII.A.2 QIII.A.3 QIII.B.1 QIII.B.2 QIII.B.3 QIII.B.4 QIII.B.5 QIII.C.1 QIII.C.2 QIII.C.3 QIII.C.4 QIII.C.5 QIII.C.6